What
is a number? If we could see them, what might the set of natural
numbers look like?
You
might picture an endless line of nodes (numbers) connected by one-way
arrows, starting at 0. Next, is 1, then 2 and so on.

(For reasons that will be
apparent below, I call this the main sequence.)

From
this diagram, we can see that:

1. 0 is a number
(the first number here).

2. Every number
has a unique next number.

3. No two numbers have the same next
number.

4. No number has 0 as its next number (0
is the first number).

5. No non-empty, proper
subset of the natural numbers is completely disconnected from the remaining
numbers.

It
all seems so obvious, so self-evident, but... so what? The thing is,
this short list of “obvious and self-evident” properties
characterize the set of natural numbers to the extent that from these
properties alone, we can (in theory) derive all of number theory,
algebra and calculus. Yes, at its base, mathematics is that simple!

Let's
now translate this list of properties into the language of DC Proof.

Again, no non-empty,
proper subset of the natural numbers is completely disconnected from the
remaining numbers.Formal Proof (149
lines).

It
can be shown that if set n',
element 0'and
function next'satisfy
the above axioms (with the obvious substitutions in the above axioms), then nand n'would
have to be
an identical “copy” of one another. The structures of n
and n'
will be identical, only the names will have been changed, i.e. n
would be
order-isomorphic
to n'. (This is not true of algebraic structures in general.)

Informally, we
can match up the elements of n
and n'
quite naturally
as follows:

0 ↔ 0'

next(0) ↔ next'(0')

next(next(0)) ↔ next'(next'(0'))

...and
so on.

This
matching up would be uniquely
given by the function fmapping nto n'such
that: